Throttling is a popular method of budget management for online ad auctions in which the platform modulates the participation probability of an advertiser in order to smoothly spend her budget across many auctions. In this work, we investigate the setting in which all of the advertisers simultaneously employ throttling to manage their budgets, and we do so for both first-price and second-price auctions. We analyze the structural and computational properties of the resulting equilibria. For first-price auctions, we show that a unique equilibrium always exists, is well-behaved and can be computed efficiently via tatonnement-style decentralized dynamics. In contrast, for second-price auctions, we prove that even though an equilibrium always exists, the problem of finding an equilibrium is PPAD-complete, there can be multiple equilibria, and it is NP-hard to find the revenue maximizing one. We also compare the equilibrium outcomes of throttling to those of multiplicative pacing, which is the other most popular and well-studied method of budget management. Finally, we characterize the Price of Anarchy of these equilibria for liquid welfare by showing that it is at most 2 for both first-price and second-price auctions, and demonstrating that our bound is tight.

翻译：节流是在线广告拍卖中一种流行的预算管理方法，平台通过调节广告商的参与概率，使其预算在多个拍卖中平稳支出。本文研究了所有广告商同时使用节流来管理预算的情形，并针对第一价格拍卖和第二价格拍卖进行了分析。我们考察了由此产生的均衡的结构与计算特性。对于第一价格拍卖，我们证明始终存在唯一的均衡，该均衡具有良好的性质，并且可以通过类似塔顿过程的分散式动态高效计算。相比之下，对于第二价格拍卖，我们证明尽管均衡始终存在，但寻找均衡的问题是PPAD完全的，可能存在多个均衡，且寻找最大化收益的均衡是NP难的。我们还将节流的均衡结果与另一种最流行且被广泛研究的预算管理方法——乘数出价——进行了比较。最后，我们基于流动福利分析了这些均衡的无政府代价，证明在第一价格和第二价格拍卖中该代价至多为2，并表明我们的界是紧的。